Q: Which of the following is the correct difference formula for $$\tan(x - y)$$?

$$\tan(x - y) = \dfrac{\tan x - \tan y}{1 + \tan x \tan y}$$

Q: Establish the identity: $$\cos(\theta - \dfrac{2\pi}{3}) = \cos \theta + \dfrac{1}{2}.$$

False; the correct identity is $$\sin(\theta - \dfrac{2\pi}{3}) = -\cos \theta$$

Q: Simplify $$\tan 50^\circ - \tan 20^\circ$$ using the appropriate sum or difference formula.

$$\tan 30^\circ = \dfrac{1}{\sqrt{3}}$$

Q: Which of the following is the correct sum formula for $$\tan(x + y)$$?

$$\tan(x + y) = \dfrac{\tan x + \tan y}{1 + \tan x \tan y}$$

Question 1

Which of the following is the correct sum formula for $$\sin(x + y)$$?

Accepted Answer

$$\sin(x + y) = \sin x \cos y + \cos x \sin y$$

Question 2

Find the exact value of $$\sin 75^\circ$$ using sum and difference formulas.

Accepted Answer

$$\dfrac{\sqrt{6} + \sqrt{2}}{4}$$

Question 3

Rewrite $$\sin 20^\circ \cos 40^\circ + \cos 20^\circ \sin 40^\circ as a $$single trigonometric function and find its exact value.

Accepted Answer

$$\sin 60^\circ = \dfrac{\sqrt{3}}{2}$$

Question 4

Given $$\sin \alpha = \dfrac{12}{13}$$ with $$\alpha in $$Quadrant I and $$\sin \beta = \dfrac{5}{13}$$ with $$\beta in $$Quadrant II, find $$\cos(\alpha + \beta).$$

Accepted Answer

$$-\dfrac{319}{481}$$

Question 5

Which of the following is the correct difference formula for $$	an(x - y)$$?

Accepted Answer

$$	an(x - y) = \dfrac{	an x - 	an y}{1 + 	an x 	an y}$$

Question 6

Find the exact value of $$	an 105^\circ$$ using sum and difference formulas.

Accepted Answer

$$-2 - \sqrt{3}$$

Question 7

Given $$\sin \alpha = \dfrac{3}{8}$$ with $$\alpha in $$Quadrant II and $$\cos \beta = \dfrac{12}{13}$$ with $$\beta in $$Quadrant IV, find $$\sin(\alpha + \beta).$$

Accepted Answer

Question 8

Establish the identity: $$\cos(	heta - \dfrac{2\pi}{3}) = \cos 	heta + \dfrac{1}{2}.$$

Accepted Answer

False; the correct identity is $$\sin(	heta - \dfrac{2\pi}{3}) = -\cos 	heta$$

Question 9

Simplify $$	an 50^\circ - 	an 20^\circ$$ using the appropriate sum or difference formula.

Accepted Answer

$$	an 30^\circ = \dfrac{1}{\sqrt{3}}$$

Question 10

Which of the following is the correct sum formula for $$\cos(x + y)$$?

Accepted Answer

$$\cos(x + y) = \cos x \cos y - \sin x \sin y$$

Question 11

Evaluate $$\cos 165^\circ$$ using sum and difference formulas.

Accepted Answer

$$-\dfrac{\sqrt{6} + \sqrt{2}}{4}$$

Question 12

Which of the following expressions is equivalent to $$\sin x \cos y + \cos x \sin y$$?

Accepted Answer

$$\sin(x + y)$$

Question 13

A sound engineer needs to calculate the exact value of $$\cos(120^\circ + 45^\circ)$$ for a phase shift in a signal. What is the value?

Accepted Answer

$$-\dfrac{\sqrt{6} + \sqrt{2}}{4}$$

Question 14

Which of the following is the correct sum formula for $$	an(x + y)$$?

Accepted Answer

$$	an(x + y) = \dfrac{	an x + 	an y}{1 + 	an x 	an y}$$

Question 15

Establish the identity: $$	an(x + y) = \dfrac{	an x + 	an y}{1 - 	an x 	an y}.$$

Accepted Answer

False; the correct formula is $$	an(x + y) = \dfrac{	an x + 	an y}{1 + 	an x 	an y}$$

Question 16

A physicist is analyzing a wave and needs to simplify $$\cos(	heta - \dfrac{2\pi}{3})$$ using sum and difference formulas. What is the simplified form?

Accepted Answer

$$\cos(	heta - \dfrac{2\pi}{3}) = -\dfrac{1}{2} \cos 	heta - \dfrac{\sqrt{3}}{2} \sin 	heta$$

Question 17

Which of the following is a valid proof step when establishing the identity $$\sin(x + y) = 	an x 	an y + \cos x \cos y$$?

Accepted Answer

This is not a valid identity; $$\sin(x + y) = \sin x \cos y + \cos x \sin y is $$correct.

Question 18

A medical imaging device uses the formula $$\cos(\alpha + \beta)$$, where $$\cos \alpha = \dfrac{3}{5}$$ and $$\cos \beta = \dfrac{5}{13}$$, both in Quadrant I. What is $$\cos(\alpha + \beta)$$?

Accepted Answer

Sum and Difference Formulas in Trigonometry: Study Notes 5.2

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